This session appears to be running under ssh. It is not recommended
to perform a upgrade over ssh currently because in case of failure it
is harder to recover.

If you continue, an additional ssh daemon will be started at port
'1022'.
Do you want to continue?

Continue [yN]<-- y

Starting additional sshd

To make recovery in case of failure easier, an additional sshd will
be started on port '1022'. If anything goes wrong with the running
ssh you can still connect to the additional one.
If you run a firewall, you may need to temporarily open this port. As
this is potentially dangerous it's not done automatically. You can
open the port with e.g.:
'iptables -I INPUT -p tcp --dport 1022 -j ACCEPT'

To continue please press [ENTER]<-- ENTER

[...]

Do you want to start the upgrade?

1 package is going to be removed. 58 new packages are going to be
installed. 587 packages are going to be upgraded.

You have to download a total of 353 M. This download will take about
1 minute with your connection.

Installing the upgrade can take several hours. Once the download has
finished, the process cannot be canceled.

Continue [yN] Details [d]<-- y

During the update, a few services might need a restart. If the update manager asks you which services should be restarted (it shows a list of services), just accept the default services by pressing ENTER.

At the end of the upgrade process, you should remove obsolete packages:

Remove obsolete packages?

18 packages are going to be removed.

Continue [yN] Details [d] <-- y

The server needs to be rebooted to complete the upgrade:

System upgrade is complete.

Restart required

To finish the upgrade, a restart is required.
If you select 'y' the system will be restarted.

Continue [yN]<-- y

After the reboot, your server is running Ubuntu 12.10. To be sure, take a look at /etc/issue: